384 XV NORMED ALGEBRAS AND SPECTRAL THEORY PROBLEMS 1. Let A be the algebra with involution ^gW °f continuous bounded complex-valued functions on R, and let /x be a bounded positive measure on R, with support equal to R. Show that is a bitrace on A for which the prehilbert space A/ng = A is separable, but that the star algebra tfg is not separable (Section 7.4, Problem 4). 2. Let A be Lebesgue measure on R and let A be the subalgebra with involution consisting of square-A-integrable functions. Then g(x, y) = J x(t)y(t) dX(t) is a bitrace on A satisfying (U) and (N), but the measure mg is not bounded, and the right-hand side of (15.9.3) is not defined for all x e A. If B c A is the subalgebra of ^£(R) consisting of A-integrable functions, then is a positive linear form on B satisfying (15.9,3) but the measure mg is not bounded. 3. Let A be the subalgebra with involution of ^C(I)» where I = [0, 1 ], consisting of func- tions x with continuous second-order derivative on I and such that x(0) = 0. Show that /(*)= Cx(t)dt + Jo is a positive linear form on A, such that the corresponding bitrace g satisfies the con- dition (U) but not the condition (N). (If (/,,) is a sequence of functions belonging to ^C(I), with values in [0, 1] and continuous second derivatives, and such that/n(0 = 1 in a neighborhood of 0 and/n(0 — 0 for t^.l/n, consider in the Hilbert space Hff the sequence of functions xn e A such that xn(t) — //„(/).) 4. Let A be the subalgebra with involution of #C(I) consisting of continuously differenti- able functions on I which vanish at 0. Show that the formf(x) — x'(0) on I is a positive linear form for which the corresponding bitrace g is zero (and therefore satisfies (U) and (N)) but does not satisfy (B). 5. Let A be the algebra with involution (the involution being x\-. >x) of complex- valued functions on [0, 1] of the form P(f, log f), where P is a polynomial in two variables with r1 complex coefficients. Then the linear form f(x) = x(t) dt is positive, and satisfies Jo (B) because A has a unit element; the corresponding bitrace g satisfies (N) but not (U). 6. Let F be a set endowed with an associative law of composition (x,y)\-+xy and a neutral element e, and let x\-+x* be an involution on F (i.e., a bijection of F onto F such that e* « e, (x*)* = x and (xy)* = y*x*). A representation of F in a Hilbertllows in all cases.